What's the correct MIME-Type for .css.br files? For ".css" it's "text/css" and for ".js.br" it's "application/x-br" (at least it's working for me). But what to use for ".css.br"?
I'm not sure the question makes sense and I am not sure what you mean when you say "at least it's working for me"?
The mime-type indicates the file type. If the file ends in .br then the file type is a brotli file - no matter whether it's a brotli compressed css or a brotli compressed js file.
It doesn't look like there is an officially registered brotli mime-type, though a quick Google shows "application/x-br" has been suggested as has application/brotli.
If you are pre-compressing files and want to send them from a web server to the client, then you should send them as .css files (not .css.br) and set the appropriate HTTP Headers:
content-encoding:br
content-type:text/css; charset=utf-8
And similarly for JavaScript files.
Precompression is just a way of saving your web server the trouble of compressing on the fly, but whether it's a precompressed file, or an on-fly compressed file shouldn't result in a difference to what is delivered to the browser.